Gravitational Potential Energy

Gravitational Potential Energy
In the position of a particle changes on account of conservative force acting on it, then the changes in its potential energy is just the amount of work done on the body by the conservative forces.

If  is the gravitational force acting on a test mass of mass m_o at distance r from the centre of earth, then



Work done in lifting the test mass through displacementby the internal agent



Total work done in lifting the test mass from r₁ to r₂ (> r₁)



Total work done by gravitational field



=  Change in potential energy

If the gravitational potential energy at infinity U₁ = 0  and  r₁ =and  r₂ = r, then



Gravitational Potential
Gravitational potential due to gravitational force of the earth is defined as the potential energy of a particle of unit mass at that point.

Potential at position

Relation Between Potential and Field

We have



In vector form

Gravitational Potential Due to Hollow Spherical Shell of Mass M, Radius R

1. Outside Point (r > R)
   
   
   
2. On the surface (r = R)
   
   
   
3. Inside the shell (r < R)
   
   

Gravitational Potential Due to Solid Sphere of Mass M, Radius R

1. Outside point (r > R)
   
   
   
2. On the surface (r = R)
   
   
   
3. Inside the sphere (r < R)
   
   
   

Gravitational Field for Uniform Solid Sphere of Mass M, Radius R

1. Outside point (r > R)
   
   
   
2. On the surface (r = R)
   
   
   
3. Inside point (r < R)
   
   

Variation of (l) with distance (r) from centre